# Questions tagged [projective-modules]

For questions about projective modules over a ring and projective objects in related categories.

146 questions
Filter by
Sorted by
Tagged with
179 views

1 vote
109 views

### Existence of a finite resolution

I have tried to formulate a question in which I was very curious, any hints suggestions are also welcomed. Thanks in advance. Let $M$ be an $R$ module ($R$ commutative ring with unity). It is given ...
1 vote
102 views

### Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...
231 views

### Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...
290 views

### In a category with a projective generator, do morphisms from the generator determine the object?

I have a cocomplete abelian category $\mathcal C$ and two objects $X$, $Y$ in $\mathcal C$. Further, $\mathcal C$ has a projective generator $P$. I have an isomorphism  \mathcal C(P,X) \cong \...
230 views

### An algebra which is a direct sum of simple sub-bimodules over a subalgebra

Let $A$ be an infinite-dimensional noncommutative algebra over a field, let $B$ be an infinite-dimensional subalgebra of $A$, and let $A$ be a direct sum of projective simple $B$-sub-bimodules. Then ...
238 views

### Are finite projective modules over $R[t]$ free when $R$ is DVR?

Let $R$ be a discrete valuation ring (DVR) and let $M$ be a projective module of finite type over the polynomial ring $R[t]$. Is $M$ free over $R[t]$? As far as I understand, this should be a ...
122 views

### Profinite projective and free modules

I am studying cohomology of profinite groups and the following question came to my mind: suppose we have $G$ a pro-$p$ group which is Poincaré Dual of dimension $d$. This means that $\mathbb{Z}_p$ as ...
139 views

### Are finitely presented algebras over VNRs projective?

Question: Let $A$ be a commutative von Neumann regular ring, and $B$ an $A$-algebra of finite presentation, i.e. $B = A[x_1, \ldots, x_n]/(f_1, \ldots, f_m)$. Is $B$ a projective $A$-module? In the ...
487 views

### Rings such that torsion-free/flat/projective modules are flat/projective/free

While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, ...
768 views

This is from Hartshrone exercise 6.6 part (a). Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following $M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{... 4 votes 2 answers 303 views ### Projective modules restricted to smooth curves I asked this question on Stack Exchange, but no one answered this. I want to prove a coherent sheaf$M$on$X$is locally free if and only if this is true for$M|_{X'}$, for all smooth curves$X'$... 3 votes 1 answer 194 views ### Is a tower of locally-free modules locally a tower of free modules? Suppose we have a (commutative, unital) ring$R$and a (commutative, unital)$R$-algebra$A$such that$A$is projective of constant rank$n$as an$R$-module. This condition is equivalent to there ... 8 votes 2 answers 406 views ### tangent bundle on noncommutative manifold Using the Serre-Swan's theorem, one can do vector bundle theory on noncommutative manifold$(A,H,D)$, by replacing vector bundle by finitely generated projectve module$M$. For the construction of ... 1 vote 2 answers 307 views ### When splitting of short exact sequence preserves the kernels This is a problem that I thought at first was obvious but that became less clear the more I thought about it. Assume we have a finitely generated algebra$A$over a field$k$, and a short exact ... 6 votes 1 answer 123 views ### On the finiteness of an Auslander-Reiten component I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2)$\Rightarrow $... -1 votes 1 answer 139 views ### infinite left degrees I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let$f: X \rightarrow Y$be an irreducible morphism ... 1 vote 1 answer 157 views ### About composition factors [closed] I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3$A$is assumed to be an Artin algebra and mod(A) the ... 1 vote 1 answer 291 views ### Question on simple modules and projective covers I have the following question: Let$A$be an Artin algebra. Let$S_1$and$S_2$be simple modules in$\text{mod}(A)$and let$P(S_1)$be the projective cover of$S_1$. Let$f: P(S_1) \rightarrow S_2$... 3 votes 2 answers 180 views ### Question on injective hulls How can I show the following: Let$f: M \rightarrow N$be a morphism in$\text{mod}(A)$, where$A$is an Artin algebra. Suppose$f \neq 0$. Then there exists a simple module$S$with its injective ... 0 votes 1 answer 320 views ### injective hull and projective cover of simple modules are indecomposable Let$A$be an Artinian algebra. Let$S$be a simple module over$A$. Let$\pi: S \rightarrow I$be the injective hull and$\tau: P \rightarrow S$be the projective cover of$S$. Then$I$and$P$must ... 6 votes 1 answer 158 views ### Finitely presented modules admitting projective covers A ring$R$is called semi-perfect if every finitely generated$R$-module has a projective cover, and it can be proved that this is equivalent to say that the category consisting of the finitely ... 0 votes 1 answer 422 views ### Separability of$\mathbb{C}[x]$over its$\mathbb{C}$-subalgebras For commutative rings$R \subseteq S$, recall that$S$is separable over$R$, if$S$is a projective$S \otimes_R S$-module, via$f: S \otimes_R S \to S$given by:$f(s_1 \otimes_R s_2)=s_1s_2$. ... 5 votes 1 answer 347 views ### Trace ideal of a projective module In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes Let$M$be a projective$R$-module... The trace of$M$is defined to be the image of the map$M \otimes_R ...
137 views

Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
1 vote
174 views

### Second summand to make projective module free

Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$? ...
353 views

### Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ... 155 views

Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $... 2 votes 1 answer 138 views ### A weak Schur's lemma for non-semisimple finite dimensional algebras Let$B \subseteq C$be an inclusion of finite dimensional (associative) algebras over a field$k$. Assume that$C$is a free$B$-module. Let$\bigoplus_i U_i$be a decomposition of$B$into ... 1 vote 0 answers 129 views ### Lifting idempotents and projective coverings --- reference request Let$R$be a (non-commutative, unital) ring with Jacobson radical$J$, write$\overline{R}=R/J$and denote the quotient map$R\to \overline{R}$by$a\mapsto \overline{a}$. It is easy to see that if ... 2 votes 1 answer 248 views ### A module associated to an endomorphism of a vector bundle Let$E$be a vector bundle over a compact connected Hausdorff space$X$. To an endomorphism$\alpha \in End(E)$, we associate a$C(X)-$module$\Gamma(E,\alpha)$consisting of all$\beta\in End(E)$... 5 votes 0 answers 144 views ### On existence of finitely generated projective generator with commutative endomorphism ring in${}_R Mod$Let$R$be a ring with unity (not necessarily commutative). Let${}_R Mod$be the category of left$R$-modules. If${}_R Mod$has a projective generator with commutative endomorphism ring , then does$...
In "Supersingular K3 surfaces" by TetsuJi Shioda, when proving Theorem 3.5 (Deligne) he considers a supersingular elliptic curve $C$ over an algebraic closed field of $\text{char}\ p>0$ and let \$R ...